Key agreement and transport protocol with implicit signatures

ABSTRACT

A key establishment protocol between a pair of correspondents includes the generation by each correspondent of respective signatures. The signatures are derived from information that is private to the correspondent and information that is public. After exchange of signatures, the integrity of exchange messages can be verified by extracting the public information contained in the signature and comparing it with information used to generate the signature. A common session key may then be generated from the pubilc and private information of respective ones of the correspondents.

This application is a divisional of application(s) application Ser. No.08/966,766 now U.S. Pat. No. 6,122,736 filed on Nov. 7, 1997.

The present invention relates to key agreement protocols for transferand authentication of encryption keys.

To retain privacy during the exchange of information it is well known toencrypt data using a key. The key must be chosen so that thecorrespondents are able to encrypt and decrypt messages but such that aninterceptor cannot determine the contents of the message.

In a secret key cryptographic protocol, the correspondents share acommon key that is secret to them. This requires the key to be agreedupon between the correspondents and for provision to be made to maintainthe secrecy of the key and provide for change of the key should theunderlying security be compromised.

Public key cryptographic protocols were first proposed in 1976 byDiffie-Hellman and utilized a public key made available to all potentialcorrespondents and a private key known only to the intended recipient.The public and private keys are related such that a message encryptedwith the public key of a recipient can be readily decrypted with theprivate key but the private key cannot be derived from the knowledge ofthe plaintext, ciphertext and public key.

Key establishment is the process by which two (or more) partiesestablish a shared secret key, called the session key. The session keyis subsequently used to achieve some cryptographic goal, such asprivacy. There are two kinds of key agreement protocol; key transportprotocols in which a key is created by one party and securelytransmitted to the second party; and key agreement protocols, in whichboth parties contribute information which jointly establish the sharedsecret key. The number of message exchanges required between the partiesis called the number of passes. A key establishment protocol is said toprovide implicit key authentication (or simply key authentication) ifone party is assured that no other part aside from a speciallyidentified second party may learn the value of the session key. Theproperty of implicit key authentication does not necessarily mean thatthe second party actually possesses the session key. A key establishmentprotocol is said to provide key confirmation if one party is assuredthat a specially identified second party actually has possession of aparticular session key. If the authentication is provided to bothparties involved in the protocol, then the key authentication is said tobe mutual; if provided to only one party, the authentication is said tobe unilateral.

There are various prior proposals which claim to provide implicit keyauthentication.

Examples include the Nyberg-Rueppel one-pass protocol and theMatsumoto-Takashima-Imai (MTI) and the Goss and Yacobi two-passprotocols for key agreement.

The prior proposals ensure that transmissions between correspondents toestablish a common key are secure and that an interloper cannot retrievethe session key and decrypt the ciphertext. In this way security forsensitive transactions such as transfer of funds is provided.

For example, the MTI/AO key agreement protocol establishes a sharedsecret K, known to the two correspondents, in the following manner.

1. During initial, one-time setup, key generation and publication isundertaken by selecting and publishing an appropriate system prime p andgenerator α of the multiplicative group Z*_(p), that is, αεZ*_(p); in amanner guaranteeing authenticity. Correspondent A selects as a long-termprivate key a random integer “a”,1<a<p−1, and computes a long-termpublic key Z_(A)=α^(Z) mod p. Correspondent B generates analogous keysb, z_(B). Correspondents A and B have access to authenticated copies ofeach other's long-term public key.

2. The protocol requires the exchange of the following messages.

A→B:α ^(x) mod p  (1)

A→B:α ^(y) mod p  (2)

where x and y are integers selected by correspondents A and Brespectively.

The values of x and y remain secure during such transmissions as it isimpractical to determine the exponent even when the value of α and theexponentiation is known provided of course that p is chosen sufficientlylarge.

3. To implement the protocol the following steps are performed each timea shared key is required.

(a) A chooses a random integer x, 1≦x≦p−2, and sends B message (1) i.e.α^(x) mod p.

(b) B chooses a random integer y, 1≦y≦p−2; and sends A message (2) i.e.α^(y) mod p.

(c) A computes the key K=(α^(y))^(a)z_(B) ^(x) mod p.

(d) B computes the key K=(α^(x))^(b)z_(A) ^(y) mod p.

(e) Both share the key K=αbx+ay.

In order to compute the key K, A must use his secret key a and therandom integer x, both of which are known only to him. Similarly B mustuse her secret key a and random integer y to compute the session key K.Provided the secret keys a,b remain uncompromised, an interloper cannotgenerate a session key identical to the other correspondent.Accordingly, any ciphertext will not be decipherable by bothcorrespondents.

As such this and related protocols have been considered satisfactory forkey establishment and resistant to conventional eavesdropping orman-in-the middle attacks.

In some circumstances it may be advantageous for an adversary to misleadone correspondent as to the true identity of the other correspondent.

In such an attack an active adversary or interloper E modifies messagesexchanged between A and B, with the result that B believes that heshares a key K with E while A believes that she shares the same key Kwith B. Even though E does not learn the value of K the misinformationas to the identity of the correspondents 5 may be useful.

A practical scenario where such an attack may be launched successfullyis the following. Suppose that B is a bank branch and A is an accountholder. Certificates are issued by the bank headquarters and within thecertificate is the account information of the holder. Suppose that theprotocol for electronic deposit of funds is to exchange a key with abank branch via a mutually authenticated key agreement. Once B hasauthenticated the transmitting entity, encrypted funds are deposited tothe account number in the certificate. If no further authentication isdone in the encrypted deposit message (which night be the case to savebandwidth) then the deposit will be made to E's account.

It is therefore an object of the present invention to provide a protocolin which the above disadvantages are obviated or mitigated.

According therefore to the present invention there is provided a methodof authenticating a pair of correspondents A,B to permit exchange ofinformation therebetween, each of said correspondents having arespective private key a,b and a public key p_(A)p_(B) derived from agenerator a and respective ones of said private keys a,b, said methodincluding the steps of

i) a first of said correspondents A selecting a first random integer xand exponentiating a function f(α) including said generator to a powerg(x) to provide a first exponentiated function f(α)^(g(x));

ii) said first correspondent A generating a first signature s_(A) fromsaid random integer x and said first exponentiated function f(α)^(g(x));

iii) said first correspondent A forwarding to a second correspondent B amessage including said first exponentiated function f(α)^(g(x)) and thesignature s_(A);

iv) said correspondent B selecting a second random integer y andexponentiating a function f′(α) including said generator to a power g(y)to provide a second exponentiated function f′(α)^(g(y)) and a signatures_(B) obtained from said second integer y and said second exponentiatedfunction f′(α)^(g(y));

v) said second correspondent B forwarding a message to said firstcorrespondent A including said second exponentiated functionf′(α)^(g(y)) and said signature s_(B).

vi) each of said correspondents verifying the integrity of messagesreceived by them by computing from said signature and said exponentiatedfunctioning such a received message a value equivalent to saidexponentiated function and comparing said computed value and saidtransmitted value;

vii) each of said correspondents A and B constructing a session key K byexponentiating information made public by said other correspondent withsaid random integer that is private to themselves.

Thus although the interloper E can substitute her public keyp_(E)=α^(ae) in the transmission as: part of the message, B will usep_(E) rather than p_(A) when authenticating the message. Accordingly thecomputed and transmitted values of the exponential functions mill notcorrespond.

Embodiments or the invention will now be described by way of exampleonly with reference to the accompanying drawings in which:

FIG. 1 is a schematic representation of a data communication system.

FIG. 2 is a flow chart illustrating the steps of authenticating thecorrespondents shown in FIG. 1 according to a first protocol.

Referring therefore to FIG. 1, a pair of correspondents, 10,12, denotedas correspondent A and correspondent B, exchange information over acommunication channel 14. A cryptographic unit 16,18 is interposedbetween each of the correspondents 10,12 and the channel 14. A key 20 isassociated with each of the cryptographic units 16,18 to convertplaintext carried between each unit 16,18 and its respectivecorrespondent 10,12 into ciphertext carried on the channel 14.

In operation, a message generated by correspondent A, 10, is encryptedby the unit 16 with the key 20 and transmitted as ciphertext overchannel 14 to the unit 18.

The key 20 operates upon the ciphertext in the unit 18 to generate aplaintext message for the correspondent B, 12. Provided the keys 20correspond, the message received by the correspondent 12 will be thatsent by the correspondent 10.

In order for the system shown in FIG. 1 to operate it is necessary forthe keys 20 to be identical and therefore a key agreement protocol isestablished that allows the transfer of information in a public mannerto establish the identical keys. A number of protocols are available forsuch key generation and are variants of the Diffie-Hellman key exchange.Their purpose is for parties A and B to establish a secret session keyK.

The system parameters for these protocols are a prime number p and agenerator α^(a) of the multiplicative group Z*_(p). Correspondent A hasprivate key a and public key p_(A)=α^(a). Correspondent B has privatekey b and b public key p_(B)=α^(b). In the protocol exemplified below,text_(A) refers to a string of information that identifies party A. Ifthe other correspondent B possesses an authentic copy of correspondentA's public key, then text_(A) will contain A's public-key certificate,issued by a trusted center; correspondent B can use his authentic copyof the trusted center's public key to verify correspondent A'scertificate, hence obtaining an authentic copy of correspondent A'spublic key.

In each example below it is assumed that, an interloper E wishes to havemessages from A identified as having originated from E herself. Toaccomplish this, E selects a random integer e, 1≦e≦p−2, computesp_(E)=(p_(A))^(e)=α^(ae) mod p, and gets this certified as her publickey. E does not know the exponent ae, although she knows e. Bysubstituting text_(E) for text_(A), the correspondent B will assume thatthe message originates from E rather than A and use E's public key togenerate the session key K. E also intercepts the message from B anduses her secret random integer e to modify its contents. A will then usethat information to generate the same session key allowing A tocommunicate with B.

To avoid interloper E convincing B that he is communicating with E, thefollowing protocol is adapted, as exemplified in FIG. 2.

The purpose of the protocol is for parties A and B to establish asession key K. The protocols exemplified are role-symmetric andnon-interactive.

The system parameters for this protocol are a prime number p and agenerator α of the multiplicative group Z*_(p). User A has private key aand public key p_(A)=α^(a). User B has private key b and public keyp_(B)=α^(b).

First Protocol

1. A picks a random integer x,1≦x≦p−2, and computes a value r_(A)=α^(x)and a signature s_(A)=x−r_(A)a mod (p−1). A sends {r_(A),s_(A),text_(A)}to B.

2. B picks a random integer y,1≦y≦p−2, and computes a value r_(B)=α^(y)and a signature s_(B)=y−r_(B)b mod (p−1). B sends {r_(B), s_(B),text_(B)} to A.

3. A computes α^(s) ^(_(B)) (p_(B))^(r) ^(_(B)) and verifies that thisis equal to r_(B). A computes the session key K=(r_(B))^(X)=α^(xy).

4. B computes α^(s) ^(_(A)) (p_(A))^(r) ^(_(A)) and verifies that thisis equal to r_(A). B computes the session key K=(r_(A))^(y)=α^(xy).

Should E replace text_(A) with text_(E), B will compute α^(s) ^(_(B))(p_(E))^(r) ^(_(A)) which will not correspond with the transmitted valueof r_(A). B will thus be alerted to the interloper E and will proceed toinitiate another session key.

One draw back of the first protocol is that it does not offer perfectforward secrecy. That is, if an adversary learns the long-term privatekey a of party A, then the adversary can deduce all of A's past sessionkeys. The property of perfect forward secrecy can be achieved bymodifying Protocol 1 in the following way.

Modified First Protocol

In step 1, A also sends α^(x) ^(₁) to B, where x₁ is a second randominteger generated by A. Similarly, in step 2 above, B also sends α^(y)^(₁) to A, where y₁ is a random integer. A and B now compute the keyK=α^(xy)⊕α^(x) ^(₁) ^(y) ^(₁) .

Another drawback of the first protocol is that if an adversary learnsthe private random integer x of A, then the adversary can deduce thelong-term private key a of party A from the equation s_(A)=x−r_(A)a {modp−1}. This drawback is primarily theoretical in nature since a welldesigned implementation of the protocol will prevent the privateintegers from being disclosed.

Second Protocol

A second protocol set out below addresses these two drawbacks.

1. A picks a random integer X,1≦x≦p−2, and computes (p_(B))^(x), α^(x)and a signature s_(A)=x+a(p_(B))^(x) {mod (p−1)}. A sends{α^(x),s_(A),text_(A)} to B.

2. B picks a random integer y,1≦y≦p−2, and computes (p_(A))^(y), α^(y)and a signature s_(B)=Y+b(p_(A))^(y) {mod (p−1)}. B sends{α^(Y),s_(B),text_(B)} to A.

3. A computes (α^(y))^(a) and verifies that α^(s) ^(_(B)) (p_(B))^(−α)^(ay) =α^(y). A then computes session key K=α^(ay)(p_(B))^(x).

4. B computes (α^(x))^(b) and verifies that α^(s) ^(_(A)) (p_(A))^(−α)^(bx) =α^(x). A then computes session key K=α^(ay)(p_(B))^(x).

The second protocol improves upon the first protocol in the sense thatif offers perfect forward secrecy. While it is still the case thatdisclosure of a private random integer x allows an adversary to learnthe private key a, this will not be a problem in practice because A candestroy x as soon as he uses it in step 1 of the protocol.

If A does not have an authenticated copy of B's public key then B has totransmit a certified copy of his key to B at the beginning of theprotocol. In this case, the second protocol is a three-pass protocol.

The quantity s_(S) serves as A's signature on the value α^(x). Thissignature has the novel property that it can only be verified by partyB. This idea can be generalized to all ElGamal-like signatures schemes.

A further protocol is available for parties A and B to establish asession key K.

Third Protocol

The system parameters for this protocol are a prime number p and agenerator α for the multiplicative group Z*_(p). User A has private keya and public key p_(A)=α^(a). User B has private key b and public keyp_(B)=α^(b).

1. A picks two random integers x,x₁,1≦x,x₁≦p−2, and computes r_(x) ₁=α^(x) ^(₁) ,r_(A)=α^(x) and (r_(A)) ^(_(x1)) , then computes asignature s_(A)=xr_(x) ₁ −(r_(A))^(r) ^(_(x1)) a mod (p−1). A sends{r_(A),s_(A), α^(x) ^(₁) , text_(A)} to B.

2. B picks two random integers y,y₁,1≦y,y₁≦p−2, and computes r_(y) ₁=α^(y) ^(₁) ,r_(B)=α^(y) and (r_(B))^(r) ^(_(y1)) , then computes asignature s_(B)=yr_(y) ₁ −(r_(B))^(r) ^(_(y1)) {mod (p−1)}. B sends{r_(B),s_(B), α^(y) ^(₁) ,text_(B)} to A.

3. A computes α^(s) ^(_(B)) (p_(B))^((r) ^(_(B)) ^()r) ^(_(y1)) andverifies that this is equal to (r_(B))^(r) ^(_(y1)) . A computes sessionkey K=(α^(y) ^(₁) )^(x) ^(₁) =α^(x) ^(₁) ^(y) ^(₁) .

4. B computes α^(s) ^(_(A)) (p_(A))^((r) ^(_(A)) ^()r) ^(_(x1)) andverifies that this is equal to (r_(A))^(r) ^(_(x1)) . B computes sessionkey K=(α^(x) ^(₁) )^(y) ^(₁) =α^(x) ^(₁) ^(y) ^(₁) .

In these protocols, (r_(A),s_(A)) can be thought of as the signature ofr_(x) ₁ , with the property that only A can sign the message r_(x) ₁ .

Key Transport Protocol

The protocols described above permit the establishment andauthentication of a session key K. It is also desirable to establish aprotocol in which permits A to transport a session key K to party B.Such a protocol is exemplified below.

1. A picks a random integer x, 1≦x≦p−2, and computes r_(A)=α^(x) and asignature s_(A)=x−r_(A)a {mod (p−1)}. A computes session keyK=(p_(B))^(x) and sends {r_(A),s_(A),text_(A)} to B.

2. B computes α^(a) ^(_(A)) (p_(A))^(r) ^(_(A)) and verifies that thisquantity is equal to r_(A). B computes session key K=(r_(A))^(b).

All one-pass key transport protocols have the following problem ofreplay. Suppose that a one-pass key transport protocol is used totransmit a session key K from A to B as well as some text encrypted withthe session key K. Suppose that E records the transmission from A to B.If E can at a later time gain access to B's decryption machine (but notthe internal contents of the machine, such as B's private key), then, byreplaying the transmission to the machine, E can recover the originaltext. (In this scenario, E does not learn the session key K.).

This replay attack can be foiled by usual methods, such as the use oftimestamps. There are, however, some practical situations when B haslimited computational resources, in which it is more suitable at thebeginning of each session for B to transmit a random bit string k to A.The session key that is used to encrypt the text is then k⊕K, i.e. kXOR'd with K.

All the protocols discussed above have been described in the setting ofthe multiplicative group Z*_(p). However, they can all be easilymodified to work in any finite group in which the discrete logarithmproblem appears intractable. Suitable choices include the multiplicativegroup of a finite field (in particular the elliptic curve defined over afinite field. In each case an appropriate generator α will be used todefine the public keys.

The protocols discussed above can also be modified in a straightforwardway to handle the situation when each user picks their own systemparameters p and a (or analogous parameters if a group other than Z*_(p)is used).

We claim:
 1. A method of authenticating a pair of correspondents A, B topermit exchange of information therebetween, each of said correspondentshaving a respective private key a, b and a public key pA, pB derivedfrom a generator α and respective ones of said private keys a, b, saidmethod including the steps of (i) a first of said correspondents Aselecting a first random integer x and exponentiating a function ƒ(α)including said generator to a power g(x) to provide a firstexponentiated function ƒ(α)^(g(x)); (ii) said first correspondent Agenerating a first signature s_(A) from said first random integer x andsaid exponentiated function ƒ(α)^(g(x)); (iii) said first correspondentA forwarding to a second correspondent B a message including said firstexponentiated function ƒ(α)^(g(x)) and said signature s_(A); (iv) saidcorrespondent B selecting a second random integer y and exponentiating afunction ƒ′(α) including said generator to a power g(y) to provide asecond exponentiated function ƒ′(α)^(g(y)) and generating a signatures_(B) obtained from said second integer y and said second exponentiatedfunction ƒ′(α)^(g(y)); (v) said second correspondent B forwarding amessage to said first correspondent A including said second exponentialfunction ƒ′(α)^(g(y)) and said signature s_(B); (vi) each of saidcorrespondents verifying the integrity of messages received by them bycomputing from said signature and said exponentiated function in such areceived message a value equivalent to said exponentiated function andcomparing said computed value and said transmitted value; (vii) each ofsaid correspondents constructing a session key K by exponentiatinginformation made public by another of said correspondents with saidfirst random integer that is private to itself.
 2. A method according toclaim 1 wherein said function ƒ(α) including said generator includes thepublic key p_(B) of said second correspondent.
 3. A method according toclaim 1 wherein said function ƒ′(α) including said generator includesthe public key p_(A) of said first correspondent.
 4. A method accordingto claim 1 wherein: said signature generated by a respective one of thecorrespondents combine the random integer, exponentiated function andprivate key of that one correspondent; and said signature ofcorrespondent A is of the form x+a(p_(B))^(x) mod(p−1).
 5. A methodaccording to claim 1 wherein: said signature generated by a respectiveone of the correspondents combine the random integer, exponentiatedfunction and private key of that one correspondent; and said signatureof correspondent A is of the form xr_(x) ₁ −(r_(A))^(r) ^(_(x1)) amod(p−1) where x₁ is a second random integer selected by A and r_(x) ₁=α^(x) ^(₁) .
 6. A method according to claim 1 wherein: said signaturegenerated by a respective one of the correspondents combine the randominteger, exponentiated function and private key of that onecorrespondent; and said signature of correspondent B is of the formy+b(p_(A))^(y) mod(p−1).
 7. A method according to claim 1 wherein: saidsignature generated by a respective one of the correspondents combinethe random integer, exponentiated function and private key of that onecorrespondent; and said signature of correspondent B is of the formyr_(y) ₁ −(r_(B))^(r) ^(_(y1)) b mod(p−1) where y₁ is a second integerselected by correspondent B and r_(y) ₁ =α^(y) ^(₁) .
 8. A methodaccording to claim 1 wherein: said signature generated by a respectiveone of the correspondents combine the random integer, exponentiatedfunction and private key of that one correspondent; said signature ofcorrespondent A is of the form x−r_(A) a mod (p−1); and saidcorrespondent A selects a second integer x₁ and forwards r_(A) ₁ tocorrespondent B where r_(A) ₁ =α^(x) ^(₁) and said correspondent Bselects a second random integer y₁ and sends r_(B) ₁ to correspondent A,where r_(B) ₁ =α^(y) ^(₁) each of said correspondents computing a pairof keys k₁,k₂ equivalent to a^(xy) and a^(x) ^(₁) ^(y) ^(₁)respectively, said session key K being generated by XORing k₁ and k₂.